Finite Group Databasebeta

← #403118

Subgroups — #403118

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2077C212yesmaximal, complemented
3C2×#1941C211yesmaximal, characteristic, complemented
4#15181C211yesmaximal, characteristic
5#15129C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11095C212yesmaximal, complemented
9#10820C211yesmaximal, characteristic, complemented
10#10721C212yesmaximal, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×C8×D4—28no—
15C2×C8×D4—24no—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21C2×#394—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28#2077—28no—
29#1941—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1442—21no—
39#1441—21no—
40#1440—22no—
41#1439—22no—
42#1438—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42C2311yescharacteristic
76C8×D4—232no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—24no—
81C2×#106—22no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8D418yes—
87C2×C4×C8—24no—
88C2×C4×C8D414yes—
89#394—44no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—24no—
100#380—42no—
101#380—42no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—42no—
119#340—42no—
120C25C2×D412yes—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4SD1614yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D414yes—
147C23×C4C2×D411yescharacteristic
148C23×C4#4114yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—216no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8—48no—
166C22×C8—48no—
167C22×C8—48no—
168C22×C8—48no—
169C22×C8C2×D414yes—
170C22×C8C2×D414yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8#4114yes—
174C4×D4—28no—
175C4×D4—216no—
176C4×D4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4#4112yes—
215C2×C4⋊C4#4112yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235#106—28no—
236#106—44no—
237C4⋊C8—28no—
238C22⋊C8—216no—
239C4×C8—28no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—81no—
249#95—81no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24#12114yes—
263C24C2×SD1614yes—
264C24#13514yes—
265C24C22×D411yescharacteristic
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—216no—
269C2×D4—216no—
270C2×D4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4#12114yes—
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4C2×SD1612yes—
351C22×C4#13512yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+412yes—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—432no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8#12014yes—
379C2×C8#12414yes—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C22⋊C4—216no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—22no—
411C42—22no—
412C42—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23#46518yes—
448C23C2×#12112yes—
449C23C2×#12111yescharacteristic
450C23#54611yescharacteristic
451C23#54611yescharacteristic
452C23#54712yes—
453C23C22×SD1611yescharacteristic
454C23C2×#13512yes—
455C23C2×#13511yescharacteristic
456C23#57611yescharacteristic
457C23#57712yes—
458C23#57811yescharacteristic
459D4—232no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#46514yes—
521C2×C4C2×#12011yescharacteristic
522C2×C4#52511yescharacteristic
523C2×C4#52511yescharacteristic
524C2×C4#52512yes—
525C2×C4C2×#12411yescharacteristic
526C2×C4#52912yes—
527C2×C4#52912yes—
528C2×C4#53511yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#53912yes—
531C2×C4#53912yes—
532C2×C4#54612yes—
533C2×C4#54612yes—
534C2×C4#54712yes—
535C2×C4#54712yes—
536C2×C4#55311yescharacteristic
537C2×C4#55311yescharacteristic
538C2×C4#55612yes—
539C8—24no—
540C8—24no—
541C8—44no—
542C22—28no—
543C22—28no—
544C22—28no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22#242512yes—
558C22#242512yes—
559C22#253511yescharacteristic
560C22#254711yescharacteristic
561C22#284711yescharacteristic
562C22C2×#46511yescharacteristic
563C22C2×#46512yes—
564C22#305011yescharacteristic
565C22#305012yes—
566C22#305711yescharacteristic
567C22#306111yescharacteristic
568C22#306212yes—
569C22#306312yes—
570C22#329511yescharacteristic
571C22#329612yes—
572C22#329811yescharacteristic
573C22#330112yes—
574C22#330211yescharacteristic
575C22#330511yescharacteristic
576C22#333512yes—
577C22#333612yes—
578C22#333912yes—
579C22#334012yes—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—162no—
586C4—162no—
587C4—161no—
588C4—161no—
589C4#239811yescharacteristic
590C4#240611yescharacteristic
591C4#243511yescharacteristic
592C4#246612yes—
593C4#246612yes—
594C4#263311yescharacteristic
595C2—48no—
596C2#1486811yescharacteristic
597C2#2523312yes—
598C2#2523512yes—
599C2#2523612yes—
600C2#2523812yes—
601C2#2814111yescharacteristic
602C2#2814411yescharacteristic
603C2#2870611yescharacteristic
604C2#2870811yescharacteristic
605C2#3292911yescharacteristic
606C2#3293811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2077C2semidirect
3. C2×#1941C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
8. #11095C2semidirect
9. #10820C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C8×D42
C2×#20772
C2×#19412
#151812
#151292
#112282
#112272
#110952
#108202
#107212